TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The kernel-cokernel sequence of a composite

Statement

For composable morphisms $A \overset{f}{\to} B \overset{g}{\to} C$ in an abelian category, there is an exact sequence $0 \to \ker (f) \to \ker (g f) \to \ker (g) \overset{q_{f} k_{g}}{\to} coker (f) \to coker (g f) \to coker (g) \to 0,$ where $k_{g} : \ker (g) \to B$ is a kernel of $g$ , $q_{f} : B \to coker (f)$ is a cokernel of $f$ , and the unlabeled arrows are the canonical comparison maps induced by the chosen kernels and cokernels.

Facts & Assumptions

Given: Composable morphisms $A \overset{f}{\to} B \overset{g}{\to} C$ .

[L1]

Under the stated endpoint hypotheses, the induced kernel and cokernel sequences are exact (Exactness of kernel and cokernel sequences under endpoint hypotheses).

[L2]

Exactness is self-dual (Exactness is self-dual).

[L3]

Kernels and cokernels are universal for the morphisms they annihilate (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers).

[L4]

The identity of an object is a kernel of its map to $0$ , and dually a cokernel of the map $0 \to A$ (The cokernel of the zero map out of the zero object is the target, and dually for kernels).

[L5]

Every kernel is monic (Every equalizer is a monomorphism, and every coequalizer is an epimorphism).

Proof

technique · direct

1.1

Choose kernels $k_{f} : K_{f} \to A, k_{g f} : K_{g f} \to A, k_{g} : K_{g} \to B$ and cokernels $q_{f} : B \to Q_{f}, q_{g f} : C \to Q_{g f}, q_{g} : C \to Q_{g} .$ Because $g f k_{f} = 0$ , [L3] gives a canonical map $u : K_{f} \to K_{g f}$ with $k_{g f} u = k_{f}$ . Likewise $g f k_{g f} = 0$ gives a canonical map $v : K_{g f} \to K_{g}$ with $k_{g} v = f k_{g f}$ . Put $δ : = q_{f} k_{g} : K_{g} \to Q_{f} .$ Since $q_{g f} g f = 0$ and $q_{g} g = 0$ , [L3] also gives canonical maps $w : Q_{f} \to Q_{g f}$ and $x : Q_{g f} \to Q_{g}$ .

L3construct

2.1

The map $u$ is monic: if $u s = u t$ , then $k_{f} s = k_{g f} u s = k_{g f} u t = k_{f} t$ , and monicity of $k_{f}$ from [L5] forces $s = t$ . So the sequence is exact at $\ker (f)$ .

L5step 1.1algebra

2.2

Apply [L1] to the commutative diagram tikzcd \ker(f) \arrow[r, "k_f"] \arrow[d, "0"'] & A \arrow[r, "f"] \arrow[d, "g f"'] & B \arrow[d, "g"'] \\ 0 \arrow[r] & C \arrow[r, "1_C"'] & C. The top row is exact, and $0 \to C$ is monic. Hence the induced sequence $\ker (0) \to \ker (g f) \to \ker (g)$ is exact. By [L4], $\ker (0)$ is represented by $1_{\ker (f)}$ , so this is exactly the sequence $\ker (f) \overset{u}{\to} \ker (g f) \overset{v}{\to} \ker (g)$ at $\ker (g f)$ .

L1L4step 1.1construct

2.3

Apply [L1] again to tikzcd A \arrow[r, "f"] \arrow[d, "g f"'] & B \arrow[r, "q_f"] \arrow[d, "g"'] & \operatorname{coker}(f) \arrow[d, "0"'] \\ C \arrow[r, "1_C"'] & C \arrow[r] & 0. The top row is exact, and $1_{C}$ is monic. Therefore the induced sequence $\ker (g f) \to \ker (g) \to \ker (0_{coker (f), 0})$ is exact. By [L4], the last kernel is represented by $1_{coker (f)}$ , and the induced map is $q_{f} k_{g} = δ$ . Hence $\ker (g f) \overset{v}{\to} \ker (g) \overset{δ}{\to} coker (f)$ is exact at $\ker (g)$ .

L1L4step 1.1construct

3.1

Apply steps 2.1 to 2.3 in the opposite category to the composable pair $C \overset{g^{o p}}{\to} B \overset{f^{o p}}{\to} A .$ Using [L2], the resulting exactness statements transport back to exactness of $\ker (g) \overset{δ}{\to} coker (f) \overset{w}{\to} coker (g f) \overset{x}{\to} coker (g) \to 0$ at $coker (f)$ , at $coker (g f)$ , and at $coker (g)$ .

L2step 1.1step 2.1step 2.2step 2.3

4.1

Steps 2.1 to 2.3 and 3.1 give the full exact sequence displayed in the statement.

step 2.1step 2.2step 2.3step 3.1∎

Depends on

Used by

The kernel-cokernel sequence of a composite of module maps Example

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Saunders Mac Lane, Categories for the Working Mathematician, Exercise VIII.4.6 (standard reference, not scraped)